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Gerd Faltings Proves Mordell's Conjecture (1983)

Gerd Faltings Proves Mordell's Conjecture (1983)

Overview

German mathematician Gerd Faltings (1954- ) proved Mordell's conjecture in 1983, an accomplishment that earned him the prestigious Fields Medal, mathematics' highest honor. His method of altering a familiar geometric theorem into algebraic terms led him to solve the complex geometric theorem proposed by Louis Mordell in 1922. His success in proving this conjecture has contributed to the advancement of the studies of algebra and geometry.

Background

Faltings has been awarded many honors in his lifetime. Most notable is his receipt of the distinguished Fields Medal, which he received in 1986. Faltings earned this honor because he proved Mordell's conjecture using algebraic geometry. The conjecture that Louis Mordell (1888-1972) initiated in 1922 stated that a given set of algebraic equations with rational coefficients defining an algebraic curve of n greater than or equal to 2 must have only a finite number of rational solutions. To prove the conjecture, Faltings used a method initiated out of developing the Arakelov theorem in order to produce an arithmetic version of yet another theorem—the commonly used, geometrically based Riemann-Roch theorem.

Mordell in his era proved the finite generation in mathematics, and mathematicians have since built upon his findings and developed the crossover use of algebra in solving geometric problems utilizing the methods of one to enable proof of the other. André Weil (1906-1998), for example, extended Mordell's result of finiteness using algebraic geometry to prove finiteness in Abelian varieties over number fields using multiple geometric strategies. Geometrically, Mordell's conjecture translated well into algebraic terms, providing scholars with the opportunity to use functional algebraic methods to solve geometric problems. Mordell's conjecture, when translated algebraically, revealed that a group of curves based on his conjecture would result in only a finite number of sections if the group were not constant. Thus, Faltings was able to prove the accuracy of the conjecture.

Impact

Faltings's proof is important for several reasons, which are listed below and then discussed in greater detail in the paragraphs following:

  1. It shows how the work of one mathematician can lead others to advance the knowledge of a given field.
  2. It has led others to find new ways to solve age-old problems related to number theory.
  3. It was instrumental in helping Andrew Wiles finally prove what was probably mathematics' greatest unsolved problem: Fermat's last theorem.

The historical context of Faltings's achievement is important because it sheds light on how the work of one mathematician can lead other scholars to advance knowledge in a given area. A mathematician named Gillet-Soule, for instance, expanded on the technique of utilizing many arithmetic surfaces to translate another well-known theorem, the Reimann-Roch theorem. Gillet-Soule also added varieties pertaining to arbitrary dimensions, complex differential geometry as it relates to the components at infinity, and real partial differential equations. He utilized this expansion of the Reimann-Roch theorem by using both the Hirzebruch-Grothendieck theorem and the original Riemann-Roch theorem. Another example of how scholars have built upon the accomplishments of others outside of their specialty in mathematics can be seen in Bismut's work. Bismut took Gillet-Soule's developments, which sprung from Faltings's work as it related specifically to real partial differential equations, and expanded them to function with the analogues of Green functions, particularly in higher dimensional cases. Following Faltings's proof of Mordell's conjecture, another mathematician—a man named P. Vojta—was later able to find another way to prove Mordell's conjecture. To accomplish this, he utilized the fundamental information that proved the Reimann-Roch theorem, and then he expounded upon particular aspects of it to create yet another way to prove Mordell's conjecture.

All of these various advancements from great scholars such as Faltings encompass the expansion of several areas of mathematics that have found their residence in the Riemann-Roch theorem. Throughout time, the Riemann-Roch theorem has been transformed into something that encompasses other arithmetic areas of study as a result of dedicated scholars who have contributed to the discovery of new elements in algebra and geometry, adding new dimensions to original theorems. Faltings has written a book (Lectures on the Arithmetic Riemann-Roch Theorem) discussing the Riemann-Roch theorem in algebraic terms.

The widespread use of contemporary methods to solve mathematical problems posed ages ago relative to number theory, as is the case with Faltings's proof of Mordell's conjecture, has led many scholars to discover new ways to incorporate several areas of mathematics to solve old problems. In his case, Faltings was able to utilize the general concepts of algebraic geometry along with complete objects, including the components of Arakelov's infinity. In using the knowledge of Arakelov's infinity theorem, Faltings illustrated the importance of building upon new discoveries. The Arakelov theory prepared the means for other mathematicians to create unification of intersection theory in algebraic fields as well as unification in classical intersection theory. Arakelov defined the infinity of intersection numbers as the values specifically of Green's functions.

Proving Mordell's conjecture led Faltings to further the study of Fermat's last theorem, long considered the greatest unsolved problem in mathematics until it was finally proven by Andrew Wiles (1953- ) in 1994. Faltings illustrated that Fermat's last theorem, xn + yn = zn, is only capable of a finite number of rational solutions in integers for which n is greater than 2 and, therefore, has no solutions. Faltings also used ideas of Vojta's to prove a conjecture created by Serge Lang that dealt with higher dimensional diophantine analogues for use in basic sub-varieties of Abelian varieties. Diophantine approximations are the formal relationships existing in the functions of a theory and the counting processes used to determine the number of functions in classical as well as specific asymptotic estimates.

Faltings's methods of algebraic geometry have also been used to examine the particular finiteness conjecture in Galois representations. The practicality of utilizing various areas in mathematics has given rise to many developments in the various fields. Interest in mathematics has grown tremendously with the advancements accomplished in number theory throughout time.

BROOKE COATES

Further Reading

Faltings, Gerd. Lectures on the Arithmetic Riemann-RochTheorem (Anal of Mathematics Studies, 127). Princeton, NJ: Princeton University Press, 1992.

Faltings, Gerd. Rational Points. Friedrich Vieweg & Sohn, 1992.

Soule, C., D. Abramovich, J. K. Kramer, and J. F. Burnol. Lectures on Arakelov Geometry (Cambridge Studies in Advanced Mathematics, 33). New York: Cambridge University Press, 1995.

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